It may seem unusual to have such low values for masses of
neutrinos, when all other particles like electrons, quarks, etc are much
heavier, with their masses relatively closely grouped. Given that particles get mass via the Higgs
mechanism, why, for example, should the electron neutrino be 105
times or more lighter than the electron, up and down quarks. That is, why would the coupling to the Higgs
field be so many orders of magnitude less?

One might not be too surprised if the Higgs coupling were
zero, giving rise to zero mass. One
might likewise not be too surprised if the coupling resulted in masses on the
order of the Higgs, or even the GUT, symmetry breaking scale.

Consider the quite reasonable possibility that after
symmetry breaking, two types of neutrino exist, with one having zero mass (no
Higgs coupling) and the other having (large) mass of the symmetry breaking
scale. As we will see, it turns out that
reasonable superpositions of these fields can result in light neutrinos (like
those observed) and a very heavy neutrino (of symmetry breaking scale, and
unobserved).

and we get the upper left
diagonal term almost 3 orders of magnitude smaller than the lower right term,
which is approximately the same as the original such term. The off diagonal terms equal the geometric
mean of the diagonal terms, i.e., ,
and are not as small as the upper left term, but significantly smaller that the
lower right one.

The fundamental point is that by starting with a matrix
of form like (1),
and transforming to another basis, which is rotated by a small angle from the
original, we get a matrix of form like (3).

We don’t know a great deal, experimentally, about
neutrino mass, but on general theoretical grounds, two distinct classes of
neutrino mass terms are allowed in the Lagrangian of electroweak
interactions. These are called Dirac and
Majorana mass terms.

Note that Majorana mass terms have nothing to do with the
Majorana representation in spinor space.
One can use any representation for the fields of which Majorana and
Dirac mass terms are composed. Neither do Majorana mass terms imply the
associated particles/fields are Majorana fermions, of which you may have heard.
Majorana fermions are their own anti-particles. More on this in Sect. 5. For now, we will assume that both Dirac and Majorana
mass terms contain only Dirac type particles (in any representation we like.)

The Dirac mass
terms, which are the usual terms dealt with in introductory quantum field
theory (QFT), have form

where sub/superscripts L
and R designate left or right hand
chirality, and the superscript c
represents charge conjugation. That is,

destroys
a LH chiral neutrino and creates a RH antineutrino,

creates “
“ “
“ and
destroys “ “ “
,

creates
“ “ “
“ and destroys “
“ “ (does same as ),

destroys “
“ “ “
and creates “
“ “ (does same as ),

and for R subscript, interchange L↔R
everywhere above.

Note that the subscript always refers to particles. For a non conjugated field, no overbar means
destroys particles, overbar means creates particles, and antiparticle actions
for the same field are just reversed from particle actions (particle ↔ antiparticle, LH ↔ RH, destroy ↔ create).

Charge conjugating a field has the same effect on
particle/antiparticle and creation/destruction as an overbar (overbar is
effectively a complex conjugate transpose [plus a γ0
multiplication]). That is, the overbar
and the superscript “c” have the same
effect. The charge conjugation merely
lets us have the overbar (row) operator effect in a non overbar (column)
vector. In fact, the symbol is used by some for the term of (5), with
similar changes for other terms, where one must keep in mind for such notation that
inner product in spinor space is implied, even though there is no obvious transpose
term (row vector on left) in .

Note that the first term in (4)
destroys a RH particle and creates a LH one.
The Feynman diagram for this term shows a RH particle disappearing at a
point and a LH particle appearing. Thus
weak (chiral) charge is not conserved, as a LH neutrino has +1/2 weak charge
and a RH neutrino has zero weak charge. Lepton number, however, is conserved,
as we started with a neutrino (not an anti-neutrino) and ended up with a
neutrino.

Somewhat similarly, the first term in (5)
creates two LH neutrinos out of the vacuum and thus also does not conserve weak
charge. But, importantly, it does not
conserve lepton number (which the Dirac terms do.) We started with zero
neutrinos and ended up with two neutrinos.

Suppose, as suggested earlier, that Higg’s or GUT
symmetry breaking only gave Majorana mass to neutrinos. That is, coupling to the Higgs (or Higges)
was not done in a way that led to Dirac mass terms. So, the mass matrix would be diagonal, unlike
(8),
of form

where we have represented the fields directly coupled to the
Higgs by (νN)T. In other words, ν and N
are the mass eigenstates for our neutrinos.

On the other hand, the weak eigenstates νL
andνR
(and their conjugates) of (7), which are linear
superpositions of ν and N,
interact directly via the weak force, and represent what we detect in weak
interaction experiments (ignoring in this context the fact that νR
has zero weak charge and does not so interact.)

Finding (10)
from (8)
is just an eigenvalue problem, with mν
and M the eigenvalues. That is, we could think of our fields in two
different, but essentially equivalent, ways: 1) a mix of Majorana and Dirac
mass terms with the column vector of fields in (7), or
2) pure Majorana mass terms associated with the mass matrix of (10),
whose associated fields are represented by the different column vector (νN)T.

Heuristically, finding (νN)T from (νLcνR)T
can be thought of as “rotating” our basis vectors in an abstract space until we
find an alignment giving the fields vector the components (νN)T.

Assuming that is the case in the real world (we have no
way of knowing via experiments to date), what would the mass matrix (10)
look like in order to give us the kind of masses (either mD or perhaps ) that we see?
Remember we are looking for a reason why neutrino mass is so much lower
than that of other particles.

That reason posits that the field components of the
vector in (11)
are the ones directly coupled to the Higgs field. It works best if the mass mν
= 0, as that means there is no Higgs coupling for the ν field, but there is such coupling for the N. (And (10)
then becomes the analog of (1) in Section 2.) Note that if
we took mν ≠ 0, but mν
<< M, we would still be left
with our original problem, which is “why is one mass so much smaller than the
others?”. Having zero mass is easier to
explain (no coupling) than extremely low mass (extremely small coupling.)

Given the treatment of Section 2, we can immediately draw conclusions about the
magnitudes of the four components of (8),
given (10)
with the upper left component equal to zero and the (νN)T basis being
close to the (νLcνR)T
basis. That is, we have the mass
hierarchy we need,

, (12)

where the Dirac mass mD is the geometric mean
of the left and right Majorana masses, the diagonal components of (8). That is,

The simple “deduce by analogy” method of the prior
section allows us to see, relatively easily, the essence of the see-saw
mechanism. But to fully quantify it, we
need the following more rigorous analysis.

The characteristic equation for the eigenvalue problem
solution of (8)
is

Some care is needed to note that the top component here
is really the νLc
field with the fractional factor indicating the size of the νLc
field compared to the νR
field. That is, N is really a superposition of the two fields, such that if νR
has a coefficient of one in that superposition, then the νLc field has a
coefficient of . In other words, in (18),
the symbol νR
really stands for the coefficient (effectively, the magnitude) of the νR
field, not the field itself (which the location in the column vector denotes.)

Note also that, up to here, we have ignored the Hermitian
conjugate half of (7),
which we will have to include. So our
true N will also include that, and
is, in terms of the fields themselves, rather than as a two component vector,
expressed as

then N is composed almost entirely of νR (and its similar
sibling νRc),
from (17)
and (23)
is very heavy, and is thus effectively sterile.
Conversely, νR
can be thought of as composed almost entirely of N. Similarly, ν is composed almost entirely of νL
(and νLc),
and conversely, νL
is almost entirely composed of the weightless ν.

From (16), one sees that for a given
value of mD,a higher value for means a lower the value for ,
and vice versa, and thus, the name “see-saw mechanism”. Further, from (21)
and (22),
the higher the value for ,
the more νR
→ N and νL → ν.

Approached in a different way, given and ,
mD will be the geometric
mean of those two masses, and will generally be closer to the lower of the
two. (If and ,
then mD = 10.) Further, if (23)
holds, from (16),
we have

An astute reader, who hadn’t read Sections 2 and 4.1, might question if we have gained anything. We originally sought a reason why the known
Dirac mass mD is so small
compared to other masses. We got that
via the eigenvalues analysis above, but in the process, we had to make another,
seemingly arbitrary, assumption (23). With this assumption, we appear merely to be
substituting one mass hierarchy problem for another. That is, we now have to ask why mD turns out to be so much
smaller than .

The answer is this.
If we start with the mass matrix (10)
with one field having zero mass (uncoupled to Higgs particle(s)),

and do a slight “rotation” in the
2D space of (νN)T,
we end up with a matrix like (8) with the characteristic (23),
which served as our initial assumption,
but which is justified if we started with (27). Our assumption boils down simply to assuming
a small transformation.

The adjective “Majorana” is applied to three distinctly
different things, which we need to distinguish between.

The first use most people see of this term is for one of
three representations of Dirac matrices and spinors. The three representations
are Dirac-Pauli (the Standard Rep), Weyl, and Majorana. As noted at the
beginning, this use of “Majorana” has nothing to do with the Majorana mass
terms of this article. Everything in this article can be done in any one of the
three representations.

Herein, we so far have been dealing with the second use
of the term regarding Majorana vs. Dirac type mass terms in the Lagrangian,
i.e., (4)
and (5).
The neutrinos and Dirac matrices in these terms can be represented by any one
of the three representations above.

The third use of the term refers to type of neutrino. A Majorana
particle is defined as a particle
that is its own antiparticle. A Dirac
particle, on the other hand, has an antiparticle that is distinctly different
from it. Typically, in almost all of one’s study of QFT, one deals with Dirac
type particles.

Neutrinos are the only particles that can be either Dirac
or Majorana types. All other fermions are known, from experiment, to be Dirac
fermions. No experiments to date (Jan 2012) have been able to determine if
neutrinos are Majorana or Dirac particles. Double beta decay experiments may
one day be able to do this.

As an aside, Majorana particles are easiest to handle
mathematically in the Majorana representation.

Note that the neutrinos we deal with in our mass terms
can be either Dirac or Majorana neutrinos, but both type mass terms would need
to involve the same particle type. From (4) and
(5),
we see that the particles in each type term are represented by the same
symbols, i.e., they represent the same
particle type (Dirac or Majorana) in both
type mass terms (Dirac and Majorana).

In summary, we can
have

·Majorana representation in spinor space (it or
one of other 2 reps can be used for any of below)

·Majorana vs Dirac mass terms in Lagrangian (both
together can be used with either particle type below)

·Majorana vs Dirac type particles (Majorana is
its own antiparticle)

6Comments on Lepton Number Conservation

With regard to Majorana vs. Dirac type mass terms in the
Lagrangian, we saw (see Wholeness Chart 1, pg. 3) how both types of terms do not conserve weak charge.
We also saw that the Majorana mass terms lead to non-conservation of lepton
number, whereas the Dirac mass terms lead to conservation of lepton number.
These results were specifically for Dirac neutrinos in both types of mass term,
where Dirac neutrinos have a lepton number +1, and Dirac antineutrinos have a
lepton number of 1.

However, what if the neutrinos we are dealing with in
experiment are actually Majorana neutrinos? Then neutrinos and anti-neutrinos
would have the same lepton number, since they are the same particle. But this
number would have to be its own negative, since quantum numbers for
anti-particles have opposite sign of those for particles. Zero is the only
number that works, so we could conclude that Majorana particles have lepton
number zero.

Therefore, for Majorana neutrinos in both types of mass
terms, all interactions solely from mass terms of either form will result in no
change of lepton number. So, if we are dealing with Majorana neutrinos, the “No”
we have in the last row, last column of Wholeness Chart 1 will change to a
“Yes”. Prior to this, we had been assuming we were working with Dirac
neutrinos.

where what we usually consider a
Dirac anti-neutrino with lepton number 1,
is now a Majorana neutrino with lepton number 0. Thus, we started with a
neutron having zero lepton number, but end up with products having a net +1
lepton number (from the electron in (28)). We
conclude that even though Majorana neutrinos in the Lagrangian mass terms (both
Dirac and Majorana mass terms) will not lead to lepton number violation,
interactions of Majorana neutrinos will.

Thus, we will have lepton number non-conservation for i) Dirac
neutrinos if, and only if, Majorana mass terms exist in the Lagrangian or ii)
Majorana fermions regardless of what mass terms are in the Lagrangian.

7Possible Physical Scenarios

There are three possible scenarios, assuming both
neutrino types exist.

Possibilities
for both Dirac and Majorana neutrinos existing in nature

1) Dirac
and Majorana fermions both interact weakly, and what we see in experiments is a
blend of both. (Not considered likely by most.)

2) Only
Dirac neutrinos interact weakly, and we don’t ever see Majorana neutrinos in
any experiments.

3) Only
Majorana neutrinos interact weakly, and we don’t ever see Dirac neutrinos in
any experiments.

Possibilities if only one type exists in nature

4) Dirac neutrinos exist, but no Majorana ones.

5) Majorana neutrinos exist, but no Dirac ones.

If the See-Saw Mechanism is
True

If the see-saw mechanism exists, then we have both type
mass terms of (4)
and (5),
and with

, (29)

and for which we could have, in
one scenario, Dirac neutrinos represented by νL and νR
if only Dirac neutrinos exist. Alternatively, we could instead have Majorana
neutrinos represented by those symbols. In either case, our interaction terms
would include the symbols νL
and νR,
along with intermediate vector boson fields.

For mD
much larger than ,
the mD mass term would not
play a role in the theory at energy levels of the present day. So we would
effectively see neutrinos, be they Dirac or Majorana neutrinos, as having mass ,
i.e., as having mass of the Majorana mass terms in .

8Summary of See-Saw Mechanism

See-saw MechanismTheory

The (common textbook) treatment covered in Section 4.2 began with a general, non-diagonal mass matrix,
looked at finding the mass eigenvalues of that matrix, and examined the
relationships engendered between the masses.
However, looking at it somewhat in reverse, as in Section 4.1, can be helpful pedagogically.

That is, start with the mass eigenstates fields ν and N,
the ones coupled directly to the Higgs field (with ν having zero coupling), and the diagonal mass
matrix (27). The weak eigenstates fields νL
and νR
(and their charge conjugation fields) are superpositions of the ν and N
fields.

We then ask “If we transform (νN)T into the (νLcνR)T,
what would the transformed mass matrix look like?” Well, if nature has chosen to make this a
slight transformation (a small “rotation” in the 2D space of the fields), which
is reasonable, then we would get a mass matrix with a very small upper left
diagonal term ,
a very large lower right diagonal term ,
and off diagonal terms mD which
are each the geometric mean of the diagonal ones, as in (16). We would have a “see-saw” relation between
the masses, and could readily have neutrino masses of the order observed. For a greater “rotation” in 2D fields space,
the greater would be the “see-saw” effect (bigger and lower ), and also the greater the value of mD.

The neutrinos we see in experiments could be either Dirac
or Majorana types, though either type would have both forms for Majorana and
Dirac mass terms in the Lagrangian.